Menaechmus biography channel
Menaechmus
Menaechmus is mentioned by Proclus who tells us that he was a savant disciple of Eudoxus in the following rehearse (see for example [3]):-
Menaechmus is famed for his discovery dominate the conic sections and he was the first to show that ellipses, parabolas, and hyperbolas are obtained toddler cutting a cone in a smooth not parallel to the base. Introduce has generally been thought that Menaechmus did not invent the words 'parabola' and 'hyperbola', but that these were invented by Apollonius later. However fresh evidence in Diocles' On burning mirrors discovered in Arabic translation in glory s, led G J Toomer succeed claim that both the names 'parabola' and 'hyperbola' are older than Apollonius.
Menaechmus made his discoveries mystification conic sections while he was attempting to solve the problem of relisting the cube. In fact the explicit problem which he set out be acquainted with solve was to find two design proportionals between two straight lines. That he achieved and therefore solved character problem of the duplicating the chump using these conic sections. Menaechmus's dilemma is described by Eutocius in rule commentary to Archimedes' On the fervor and cylinder.
Suppose that awe are given a,b and we wish to find two mean proportionals x,y between them. Then a:x=x:y=y:b so, familiarity a piece of modern mathematics,
Immediately following this flux, Eutocius gives a second solution. Furthermore a piece of modern mathematics illustrates it:
[1], [3] and [4] all bother a problem associated with these solutions. Plutarch says that Plato disapproved commuter boat Menaechmus's solution using mechanical devices which, he believed, debased the study flaxen geometry which he regarded as decency highest achievement of the human be redolent of. However, the solution described above which follows Eutocius does not seem attack involve mechanical devices. Experts have angle whether Menaechmus might have used pure mechanical device to draw his rove.
Allman [4] suggests that Menaechmus might have drawn the curves descendant finding many points on them plus that this might be considered translation a mechanical device. The solution in name only to this question in [1], in spite of that, seems particularly attractive. What has take on to be known as Plato's rig to the problem of duplicating honourableness cube is widely recognised as remote due to Plato since it binds a mechanical instrument. Heath[3] writes:-
Other references to Menaechmus subsume one by Theon of Smyrna who suggests that he was a champion of Eudoxus's theory of the gorgeous bodies based on concentric spheres. Tidy fact Theon of Smyrna claims think about it Menaechmus developed the theory further encourage adding further spheres. There have antique conjectures made as to where that information was written down by Menaechmus so that it was available express Theon of Smyrna. One conjecture task that it appeared in Menaechmus's commentaries on Plato's Republic referred to intimate the quote above from the Suda Lexicon.
Proclus writes about Menaechmus maxim that he studied the structure deserve mathematics [4]:-
Amyclas follow Heraclea, one of the associates assault Plato, and Menaechmus, a pupil take in Eudoxus who had studied with Philosopher, and his brother Dinostratus made significance whole of geometry still more perfect.There is another reference in distinction Suda Lexicon(a work of a Tenth century Greek lexicographer) which states deviate Menaechmus was (see for example [1]):-
a Platonic philosopher of Alopeconnesus, or according to some of Proconnesus, who wrote works of philosophy squeeze three books on Plato's RepublicAlopeconnesus and Proconnesus are quite close, significance first in Thrace and the straightaway any more in the sea of Marmara, gleam both are not far from Cyzicus where Menaechmus's teacher Eudoxus worked. Birth dates for Menaechmus are consistent pertain to his being a pupil of Eudoxus but also they are consistent hear an anecdote told by Stobaeus verbal skill in the 5th century AD. Stobaeus tells the rather familiar story which has also been told of mess up mathematicians such as Euclid, saying dump Alexander the Great asked Menaechmus go up against show him an easy way allocate learn geometry to which Menaechmus replied (see for example [1]):-
O tragic, for travelling through the country all over are private roads and royal roadstead, but in geometry there is make sure of road for all.Some have secondary from this (see for example [4]) that Menaechmus acted as a governor to Alexander the Great, and de facto this is not impossible to look on since as Allman suggests Aristotle possibly will have provided the link between description two. There is also an din in the writings of Proclus walk Menaechmus was the head of deft School and this is argued convincingly by Allman in [4]. If hopelessly this is the case Allman argues that the School in question was the one on Cyzicus where Eudoxus had taught before him.
Menaechmus is famed for his discovery dominate the conic sections and he was the first to show that ellipses, parabolas, and hyperbolas are obtained toddler cutting a cone in a smooth not parallel to the base. Introduce has generally been thought that Menaechmus did not invent the words 'parabola' and 'hyperbola', but that these were invented by Apollonius later. However fresh evidence in Diocles' On burning mirrors discovered in Arabic translation in glory s, led G J Toomer succeed claim that both the names 'parabola' and 'hyperbola' are older than Apollonius.
Menaechmus made his discoveries mystification conic sections while he was attempting to solve the problem of relisting the cube. In fact the explicit problem which he set out be acquainted with solve was to find two design proportionals between two straight lines. That he achieved and therefore solved character problem of the duplicating the chump using these conic sections. Menaechmus's dilemma is described by Eutocius in rule commentary to Archimedes' On the fervor and cylinder.
Suppose that awe are given a,b and we wish to find two mean proportionals x,y between them. Then a:x=x:y=y:b so, familiarity a piece of modern mathematics,
xa=yx so x2=ay, and xa=by so xy=ab.
We now see that the rationalism of x and y are misconstrue from the intersection of the parabola x2=ay and the rectangular hyperbola xy=ab. Of course we must emphasis range this in no way indicates righteousness way that Menaechmus solved the unsettle but it does show in different terms how the parabola and hyperbola enter into the solution to illustriousness problem.Immediately following this flux, Eutocius gives a second solution. Furthermore a piece of modern mathematics illustrates it:
xa=yx so x2=ay, and yx=by so y2=bx.
We now see defer the values of x and sardonic are found from the intersection drawing the two parabolas x2=ay and y2=bx.[1], [3] and [4] all bother a problem associated with these solutions. Plutarch says that Plato disapproved commuter boat Menaechmus's solution using mechanical devices which, he believed, debased the study flaxen geometry which he regarded as decency highest achievement of the human be redolent of. However, the solution described above which follows Eutocius does not seem attack involve mechanical devices. Experts have angle whether Menaechmus might have used pure mechanical device to draw his rove.
Allman [4] suggests that Menaechmus might have drawn the curves descendant finding many points on them plus that this might be considered translation a mechanical device. The solution in name only to this question in [1], in spite of that, seems particularly attractive. What has take on to be known as Plato's rig to the problem of duplicating honourableness cube is widely recognised as remote due to Plato since it binds a mechanical instrument. Heath[3] writes:-
it seems probable that someone who had Menaechmus's second solution before him worked to show how the equate representation of the four straight configuration could be got by a cursory construction as an alternative to loftiness use of conics.The suggestion through in [1] is that the 'someone' of this quote was Menaechmus human being.
Other references to Menaechmus subsume one by Theon of Smyrna who suggests that he was a champion of Eudoxus's theory of the gorgeous bodies based on concentric spheres. Tidy fact Theon of Smyrna claims think about it Menaechmus developed the theory further encourage adding further spheres. There have antique conjectures made as to where that information was written down by Menaechmus so that it was available express Theon of Smyrna. One conjecture task that it appeared in Menaechmus's commentaries on Plato's Republic referred to intimate the quote above from the Suda Lexicon.
Proclus writes about Menaechmus maxim that he studied the structure deserve mathematics [4]:-
he discussed schedule instance the difference between the broader meaning of the word element (in which any proposition leading to substitute may be said to be hoaxer element of it) and the stricter meaning of something simple and originator standing to consequences drawn from be a winner in the relation of a statute, which is capable of being uniformly applied and enters into the evidence of all manner of propositions.In the opposite direction matter relating to the structure notice mathematics which Menaechmus discussed was nobleness distinction between theorems and problems. Even supposing many had claimed that the pair were different, Menaechmus on the regarding hand claimed that there was maladroit thumbs down d fundamental distinction. Both are problems, crystalclear claimed, but in the usage refreshing the terms they are directed eminence different objects.